MECHANICS of SOLIDS ME F211 BITS Pilani K K Birla Goa Campus VIKAS CHAUDHARI Mechanics of Solids
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MECHANICS OF SOLIDS ME F211 BITS Pilani K K Birla Goa Campus VIKAS CHAUDHARI Mechanics of Solids Chapter-5 Stress-Strain-Temperature Relations Vikas Chaudhari BITS Pilani, K K Birla Goa Campus Stress-Strain-Temperature Relations Contents: Introduction Tensile test Stress- Strain diagram Idealizations of Stress-Strain Curves Elastic Stress- Strain Relations Thermal Strain Complete Equations of Elasticity Strain Energy in an Elastic Body Criteria for Initial Yielding Vikas Chaudhari BITS Pilani, K K Birla Goa Campus Stress-Strain-Temperature Relations Introduction Elastic deformation: It is the deformation which exist when load applied and it disappears as soon as the load is released. Plastic Deformation: when load is applied beyond elastic range, and the deformation does not disappears after the load is released. A Ductile materials is one which the plastic deformation is much larger than elastic deformation Vikas Chaudhari BITS Pilani, K K Birla Goa Campus Stress-Strain-Temperature Relations Tensile Test Tensile Test Displacements in Tensile Test Vikas Chaudhari BITS Pilani, K K Birla Goa Campus Stress-Strain-Temperature Relations Stress- Strain diagram Stress-strain diagram for a structural steel in tension Not to scale Draw to scale Vikas Chaudhari BITS Pilani, K K Birla Goa Campus Stress-Strain-Temperature Relations Stress- Strain diagram Typical stress- strain diagram for an aluminum alloy Offset yield point (proof stress) Vikas Chaudhari BITS Pilani, K K Birla Goa Campus Stress-Strain-Temperature Relations Stress- Strain diagram Typical stress-strain diagram for brittle material Stress-strain diagram for a cast iron in tension and compression Vikas Chaudhari BITS Pilani, K K Birla Goa Campus Stress-Strain-Temperature Relations Stress- Strain diagram Loading, Unoading and Reloading (a) Elastic behaviour Reloading of a material and (b) Partially elastic behaviour raising of the yield stress Vikas Chaudhari BITS Pilani, K K Birla Goa Campus Stress-Strain-Temperature Relations Idealizations of Stress-Strain Curves Vikas Chaudhari BITS Pilani, K K Birla Goa Campus Stress-Strain-Temperature Relations Example: Two Coaxial tubes, the inner one of 1020 CR steel and cross sectional area As, and the outer one of 2024-T4 aluminum alloy and of area Aa are compressed between heavy , flat end plates as shown in figure. We wish to determine the load-deflection curve of the assembly as it is compressed into the plastic region by an axial force P. The idealized stress strain curves are shown for both the tube materials in the second figure Vikas Chaudhari BITS Pilani, K K Birla Goa Campus Stress-Strain-Temperature Relations Solution In the range 0 0.0032, sEEEE s s s and a a a a 590 where E184 GPa s 0.0032 380 E76 GPa a 0.005 In the range 0.0032 0.005, sY s 590 MPa and a E a a E a In the range 0.005 , sY s 590 MPa and a Y a 380 MPa Equilibrium FAAPy s s s a 0 Vikas Chaudhari BITS Pilani, K K Birla Goa Campus Stress-Strain-Temperature Relations Elastic Stress- Strain Relations Assuming that the material is isotropic and homogeneous. Consider an element on which there is only one component of normal stress acting, as shown in figure. For linear elastic solid, stress is directly proportional to strain. Uniaxial tension (compression) test shows that lateral compressive (extensional) strain x is a fixed fraction of longitudinal extensional x E (compressive) strain. x This fixed fraction is known as Poisson’s ratio y z x E and given by symbol . Vikas Chaudhari BITS Pilani, K K Birla Goa Campus Stress-Strain-Temperature Relations Elastic Stress- Strain Relations For isotropic material normal stress does not produce shear strain. Similarly only normal stress y acts Analogous results are yyobtained for the strain y and x z y EEdue to z. Vikas Chaudhari BITS Pilani, K K Birla Goa Campus Stress-Strain-Temperature Relations Elastic Stress- Strain Relations For isotropic material shear stress does not produce normal strain. Each shear-stress component produces only its corresponding shear- strain component. For isotropic materials, shear stress is directly proportional to the shear strain. The proportionality constant is G (shear modulus). xy yz zx xyGGG yz zx Vikas Chaudhari BITS Pilani, K K Birla Goa Campus Stress-Strain-Temperature Relations Elastic Stress- Strain Relations 1 xE x y z 1 y y x z E 1 zE z x y xy yz zx xyGGG yz zx Relation between E, G and 11 and IEE II 21 xyGE I II E 2G 1 Vikas Chaudhari BITS Pilani, K K Birla Goa Campus Stress-Strain-Temperature Relations Thermal Strain The effect of temperature on elastic constants for many materials is small for temperature change of a hundred degree centigrade and we will not consider for our study The strain due to temperature change in the absence of stress is called thermal strain and it is denoted by superscript t on strain symbol (t) For isotropic material, thermal strain is pure expansion or contraction with no shear components. Thermal strain due to a change in temperature from To to T is t t t x y z TT o t t t 0 xy yz xz The quantity is coefficient of linear expansion. Vikas Chaudhari BITS Pilani, K K Birla Goa Campus Stress-Strain-Temperature Relations Thermal Strain Denoting the elastic strain due to stress by e and the thermal part by t, the total strain is et If material is rigidly restrained so that no strain is possible, the elastic part of the strain will be the negative of the thermal strain. Vikas Chaudhari BITS Pilani, K K Birla Goa Campus Stress-Strain-Temperature Relations Complete Equations of Elasticity Equilibrium Equations Geometric Compatibility Stress-strain Temperature Relations 1 x x y z TT o E 1 y y x z TT o E 1 z z x y TT o E xy yz zx xyGGG yz zx Vikas Chaudhari BITS Pilani, K K Birla Goa Campus Stress-Strain-Temperature Relations Example A long, thin plate of width b, thickness t, and length L is placed between two rigid walls a distance b apart and is acted on by an axial force P, as shown in figure. We wish to find the deflection of the plate parallel to the force P. Vikas Chaudhari BITS Pilani, K K Birla Goa Campus Stress-Strain-Temperature Relations Solution Assumptions 1. The axial force P results in an axial normal stress uniformly distributed over the plate area, including the end areas. 2. There is no normal stress in the thin direction (plane stress in x-y plane). 3. There is no deformation in the y direction, that is, y = 0 (plane strain in x-z plane). 4. There is no friction force at the walls. 5. The normal stress of contact between the plate and wall is uniform over the length and width of the plate. Vikas Chaudhari BITS Pilani, K K Birla Goa Campus Stress-Strain-Temperature Relations Solution Equilibrium P 0 xbt y o z xy yz xz 0 Geometric Compatibility 0 and yxL Stress Strain Relations 11 xEEE x y y y x z y x xy yz xz 0 After solving P2 PL P y x 1 z 1 bt Ebt Ebt1 L Vikas Chaudhari BITS Pilani, K K Birla Goa Campus Stress-Strain-Temperature Relations Problem The stress in a flat steel plate in a condition of plane stress are, x = 130MPa; y = -70MPa and xy = 80MPa. Find the magnitude and orientation of the principal strains in the plane of the plate. Find also the magnitudes of the third principal strain Vikas Chaudhari BITS Pilani, K K Birla Goa Campus Stress-Strain-Temperature Relations Problem In a flat steel plate which is loaded in the xy plane, it is known that -4 x = 145MPa; xy = 42MPa and z = -3.610 . What is the value of the stress y? Vikas Chaudhari BITS Pilani, K K Birla Goa Campus Stress-Strain-Temperature Relations Problem A steel pipe is held by two fixed supports as shown in figure. When mounted, the temperature of the pipe was 20oC. In use, however, cold fluid moves through the pipe, causing it to cool considerably. If we assume that the pipe has uniform temperature of -15oC and if we take the coefficient of linear expansion to be 1210-6/oC for this temperature range, determine the state of stress and strain in the central portion of the pipe as a result of this cooling. Neglect the local end effects near the supports and neglect body forces and fluid pressure and drag forces. Vikas Chaudhari BITS Pilani, K K Birla Goa Campus Stress-Strain-Temperature Relations Problem It is desired to produce a tight fit of a steel shaft in a steel pulley. The internal diameter of the hole in the pulley is 24.950mm, while the outside diameter of the shaft is 25.000mm. The pulley will be assembled on the shaft by either heating the pulley or cooling the shaft and then putting the shaft in the pulley hole and allowing the assembly to reach uniform temperature. Is it more effective to heat the pulley or to cool the shaft? What temperature change would be required in each case to produce a clearance of 0.0025 mm for easy assembly? Vikas Chaudhari BITS Pilani, K K Birla Goa Campus Stress-Strain-Temperature Relations Problem A circular bar (E = 200GPa, = 0.32, and = 11.710-6 /oC) has a diameter of 100mm. The bar is built into a rigid wall on the left, and a gap of 0.5 mm exists between the right wall and the bar prior to an increase in temperature as shown in figure.